Wonderlic Mathematics Review: Fractions, Percentages, and Related Concepts

A fraction is a number that is expressed as one integer written above another integer, with a dividing line between them (). 

It represents the quotient of the two numbers “x divided by y.” 

It can also be thought of as x out of y equal parts.

The top number of a fraction is called the numerator, and it represents the number of parts under consideration. 

The 1 in  means that 1 part out of the whole is being considered in the calculation.  The bottom number of a fraction is called the denominator, and it represents the total number of equal parts.  The 4 in  means that the whole consists of 4 equal parts. 

A fraction cannot have a denominator of zero; this is referred to as “undefined.”

Fractions can be manipulated, without changing the value of the fraction, by multiplying or dividing (but not adding or subtracting) both the numerator and denominator by the same number. 

If you divide both numbers by a common factor, you are reducing or simplifying the fraction. 

Two fractions that have the same value, but are expressed differently are known as equivalent fractions. 

For example,  are all equivalent fractions.  They can also all be reduced or simplified to .

When two fractions are manipulated so that they have the same denominator, this is known as finding a common denominator. 

The number chosen to be that common denominator should be the least common multiple of the two original denominators. 

Example:  the least common multiple of 4 and 6 is 12. 

Manipulating to achieve the common denominator: .

If two fractions have a common denominator, they can be added or subtracted simply by adding or subtracting the two numerators and retaining the same denominator. 

Example: .  If the two fractions do not already have the same denominator, one or both of them must be manipulated to achieve a common denominator before they can be added or subtracted.

Two fractions can be multiplied by multiplying the two numerators to find the new numerator and the two denominators to find the new denominator. 

Example: .

Two fractions can be divided flipping the numerator and denominator of the second fraction and then proceeding as though it were a multiplication. 

Example: .

A fraction whose denominator is greater than its numerator is known as a proper fraction, while a fraction whose numerator is greater than its denominator is known as an improper fraction.  Proper fractions have values less than one and improper fractions have values greater than one.

A mixed number is a number that contains both an integer and a fraction.  Any improper fraction can be rewritten as a mixed number. 
Example: . 

Similarly, any mixed number can be rewritten as an improper fraction. 

Example: .

Percentages can be thought of as fractions that are based on a whole of 100; that is, one whole is equal to 100%. 

The word percent means"per hundred."

Fractions can be expressed as percents by finding equivalent fractions with a denomination of 100.  Example: ; .

To express a percentage as a fraction, divide the percentage number by 100 and reduce the fraction to its simplest possible terms. 

Example:
; .

Converting decimals to percentages and percentages to decimals is as simple as moving the decimal point. 

To convert from a decimal to a percent, move the decimal point two places to the right. 

To convert from a percent to a decimal, move it two places to the left. 
Example: 0.23 = 23%; 5.34 = 534%; 0.007 = 0.7%; 700% = 7.00; 86% = 0.86; 0.15% = 0.0015.

It may be helpful to remember that the percentage number will always be larger than the equivalent decimal number.

A percentage problem can be presented three main ways:

1. Find what percentage of some number another number is.
Example: What percentage of 40 is 8?

2. Find what number is some percentage of a given number.
Example: What number is 20% of 40?

3. Find what number another number is a given percentage of.
Example: What number is 8 20% of?

The three components in all of these cases are the same: a whole (W), a part (P), and a percentage (%). 

These are related by the equation: .  This is the form of the equation you would use to solve problems of type (2). 

To solve types (1) and (3), you would use these two forms:  and .

The thing that frequently makes percentage problems difficult is that they are most often also word problems, so a large part of solving them is figuring out which quantities are what. 

Example:  In a school cafeteria, 7 students choose pizza, 9 choose hamburgers, and 4 choose tacos. 

Find the percentage that chooses tacos. 

To find the whole, you must first add all of the parts: 7 + 9 + 4 = 20. 

The percentage can then be found by dividing the part by the whole ): .

A ratio is a comparison of two quantities in a particular order. 

Example: If there are 14 computers in a lab, and the class has 20 students, there is a student to computer ratio of 20 to 14, commonly written as 20:14. 

Ratios are normally reduced to their smallest whole number representation, so 20:14 would be reduced to 10:7 by dividing both sides by 2.

A proportion is a relationship between two quantities that dictates how one changes when the other changes. 

A direct proportion describes a relationship in which a quantity increases by a set amount for every increase in the other quantity, or decreases by that same amount for every decrease in the other quantity. 

Example: Assuming a constant driving speed, the time required for a car trip increases as the distance of the trip increases.  The distance to be traveled and the time required to travel are directly proportional.

Inverse proportion is a relationship in which an increase in one quantity is accompanied by a decrease in the other, or vice versa. 
Example: the time required for a car trip decreases as the speed increases, and increases as the speed decreases, so the time required is inversely proportional to the speed of the car.